Construct the resulting probability...?

Question

The random variable is X = miles per gallon, and our interest lies in the average mileage form the inventory of the 20 cars. Construct the resulting probability distribution and what is E(X) in this distribution? Here is the info to go with the question.. You have 4 cars that get 36 mpg, 10 cars that get 50 mpg, 2 cars that get 28 mpg, and 4 cars that get 30 mpg. My question is how do you contruct a probability distribution for this?

Answer 1

X = miles per gallon

P(X=36) = 4/20 = 1/5
because you have 4 cars that get 36mpg and there are a total of 20 cars.
Similarly,
P(X=50) = 10/20 = 1/2
P(X=28) = 2/20 = 1/10
P(X=30) = 4/20 = 1/5

Check to make sure they add to one.
1/5 + 1/2 + 1/10 + 1/5
= 4/20 + 10/20 + 2/20 + 4/20
= 20/20
= 1

Now E(X) = sum of [k*P(X=k)], k = 0, 1, 2, 3, ...
For this problem, k only equals 36, 50, 28 and 30.
E(X) = 36*P(X=36) + 50*P(X=50) + 28*P(X=28) + 30*P(X=30)
E(X) = 36(1/5) + 50(1/2) + 28(1/10) + 30(1/5)
E(X) = 36/5 + 25 + 14/5 + 6
E(X) = 41